Finite Group Databasebeta

← #18528

Subgroups — #18528

Order 256.

530 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3481C211yesmaximal, characteristic
2#3152C211yesmaximal, characteristic, complemented
3#3079C212yesmaximal, complemented
4#1632C214yesmaximal, complemented
5#1630C212yesmaximal, complemented
6#1629C212yesmaximal, complemented
7#1560C211yesmaximal, characteristic, complemented
8#1489C212yesmaximal, complemented
9C2×#137—22no—
10#559C2211yescharacteristic
11#554C2212yes—
12#553—22no—
13#551—22no—
14#546—22no—
15D42C2211yescharacteristic
16#532C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic, complemented
18#529—22no—
19#525—22no—
20C4×#41C2212yes—
21C2×C4×D4—22no—
22#497—22no—
23#494C2212yescomplemented
24#492C2212yes—
25#477C2214yescomplemented
26#476C2214yes—
27#475—22no—
28#474—22no—
29#428—22no—
30#423C2212yescomplemented
31#417—22no—
32C4×M16C2211yescharacteristic, complemented
33#333C2212yescomplemented
34#331C2212yes—
35Q8⋊C8C2214yescomplemented
36D4⋊C8C2214yes—
37C2×#41D412yes—
38C2×#41D412yes—
39C22×Q8—21no—
40C22×D4—22no—
41C23×C4—22no—
42#137—48no—
43C2×Q16—24no—
44C2×SD16—24no—
45C2×M16—21no—
46C2×M16—22no—
47C4⋊Q8—22no—
48C4⋊Q8—21no—
49C4⋊Q8C2312yes—
50C4⋊Q8C2311yescharacteristic
51#127C2311yescharacteristic
52#126—24no—
53#125—22no—
54#124—22no—
55#124D412yes—
56#123—22no—
57#123—24no—
58#123—22no—
59C22⋊Q8—24no—
60C22⋊Q8—22no—
61#121—24no—
62#121—22no—
63#121—22no—
64#120—22no—
65#120—22no—
66C4×Q8D412yes—
67C4×Q8C2312yes—
68C4×D4—24no—
69C4×D4—24no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4D412yes—
76C4×D4C2312yes—
77#117—22no—
78#117—22no—
79#117D412yes—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C42—22no—
84C2×C42C2311yescharacteristic
85#107—24no—
86#106—24no—
87C4⋊C8—22no—
88C4⋊C8C2314yes—
89#103—24no—
90#102—24no—
91#97—21no—
92C4×C8C2312yes—
93C24—22no—
94#41—24no—
95#41—42no—
96#41—44no—
97C2×Q8—22no—
98C2×Q8—44no—
99C2×Q8—41no—
100C2×Q8C2×D412yes—
101C2×Q8C2×D412yes—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4C2×D412yes—
109C2×D4C2×D412yes—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—84no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120Q16—48no—
121SD16—48no—
122M16—24no—
123M16—44no—
124C2×C8—22no—
125C2×C8—24no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—24no—
132C4⋊C4—42no—
133C4⋊C4C2×D412yes—
134C4⋊C4C2×D412yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42C2×D411yescharacteristic
147C42C2×D412yes—
148C42C2411yescharacteristic
149C23—22no—
150C23—22no—
151C23—48no—
152C23—82no—
153C23#12011yescharacteristic
154Q8—22no—
155Q8—22no—
156Q8—42no—
157Q8—44no—
158Q8—82no—
159D4—22no—
160D4—42no—
161D4—44no—
162D4—48no—
163D4—42no—
164D4—42no—
165D4—82no—
166D4#13714yescomplemented
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4—82no—
183C2×C4#12012yes—
184C2×C4#12011yescharacteristic
185C2×C4C22×D411yescharacteristic
186C2×C4C22×D412yes—
187C8—24no—
188C8—44no—
189C22—22no—
190C22—24no—
191C22—21no—
192C22—42no—
193C22—42no—
194C22—44no—
195C22—42no—
196C22—88no—
197C22C2×#12011yescharacteristic
198C4—22no—
199C4—21no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—82no—
205C4#45312yes—
206C4C2×#13712yes—
207C4#57512yes—
208C2—41no—
209C2—44no—
210C2—82no—
211C2#300712yes—
212C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3481C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3152C2semidirect
3. #3079C2semidirect
4. #1632C2semidirect
5. #1630C2semidirect
6. #1629C2semidirect
7. #1560C2semidirect
8. #1489C2semidirect
17. C2×C4⋊Q8C22semidirect
23. #494C22semidirect
25. #477C22semidirect
30. #423C22semidirect
32. C4×M16C22semidirect
33. #333C22semidirect
35. Q8⋊C8C22semidirect
166. D4#137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#34812
#31522
#30792
#16322
#16302
#16292
#15602
#14892